Above, the constant \(b\)  represents the strength of external magnetic field in the spacetimeHere we use a quite different label for the strength of magnetic field, where in some other works [1]}, [2]}, [3]} is used \(B=2b\)  instead.. The transformation (REF ) leaves equations (REF ) and (REF ) unchanged for the new potentials \({\cal E}^{\prime }\)  and \(\Phi ^{\prime }\)  from which a new set of solutions \(\left\lbrace A^{\prime }_\mu ,g^{\prime }_{\mu \nu }\right\rbrace \)  to the Einstein-Maxwell equations can be extracted.
